letumsnemesislh

2022-07-03

Is there any theorem from linear algebra about writing random variables as a linear combination of their elements?
Suppose that X is a random variable which belongs to a standard probability space $\left(\mathrm{\Omega },\mathcal{B},\mu \right)$. Could anybody provide some theorem and its details where X can be written as $X=\sum _{i=1}^{k}{w}_{i}{x}_{i}$, where ${w}_{i}\ne 0$ and ${x}_{i}\in \mathbb{R}$?

billyfcash5n

Expert

Solution:
Sure, not a fancy theorem, but suppose X is a real-valued random variable with finite support $\left\{{x}_{1},...,{x}_{k}\right\}$. Then $X=\sum _{i=1}^{k}{\mathbf{1}}_{X={x}_{i}}{x}_{i}.$.
Notice the indicator weights are Bernoulli random variables.

Do you have a similar question?